A fruit punch is to be made by mixing a grape wine that is 12% alcohol, a cherry

Question

A fruit punch is to be made by mixing a grape wine that is 12% alcohol, a cherry wine that is 19% alcohol, and an apricot brandy that is 42% alcohol. All percentages are based on mass. The initial mixture has twice as much apricot brandy as grape wine, and there is 50% more cherry wine than grape wine. Your host has added to the mixture 18 kg of water and 65 kg of pure grain alcohol. This produces a quantity of punch that is 33% alcohol. How much grape wine, cherry wine, and apricot brandy were mixed together before adding the water and the alcohol? 156.7 kg grape wine 235.1 kg cherry wine 313.4 kg apricot brandy

Explanation / Answer

grape wine = x
cherry wine = y
apricot brandy = z

initial mixture has twice as much apricot brandy as grape wine means
z=2x...(i)

there is 50% more cherry wine than grape wine means
y=1.5x...(ii)

Your host has added to the mixture 18 kg of water and 65 kg of pure grain alcohol. This produces a quantity of punch that is 33% alcohol means
(12% x+19% y+42% z)+0%*18+100%*65=33% (x+y+z+18+65)

(0.12x+0.19y+0.42z)+0.00*18+1.00*65=0.33(x+y+z+18+65)...(iii)

plug (i) and (ii) into (iii)

(0.12x+0.19*1.5x+0.42*2x)+0.00*18+1.00*65=0.33(x+1.5x+2x+18+65)
(0.12x+0.285x+0.84x)+0+65=0.33(x+1.5x+2x+83)
1.245x+65=0.33x+0.33*1.5x+0.33*2x+0.33*83
1.245x+65=0.33x+0.495x+0.66x+27.39
1.245x+65=1.485x+27.39
-27.39+65=1.485x-1.245x
37.61=0.24x
x=156.7

plug value of x into (ii)
y=1.5x=1.5*156.7=235.05=235.1

plug value of x into (i)
z=2x=2*156.7=313.4

hence amount of each wine is
grape wine = 156.7 kg
cherry wine = 235.1 kg
apricot brandy = 313.4 kg

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